We first recall that hyperbolic groups have finite asymptotic dimension. This was shown initially by Roe [1]}, who confirmed an idea of Gromov from the monograph [2]}, but there are now many proofs - we use the one presented in Roe's paper [1]}, where for each \(R>0\)  he constructs a \((2R+2\delta _{m})\) -bounded cover into at most \(\vert S \vert ^{6\delta _{m}}\)  coloursthis is an incredibly bad bound that's just useful for this calculation.. In this case, the ONL number is at least \(\frac{1}{\vert S \vert ^{6\delta _{m}}}\) .
